Abstract
For a unital \(C^{*}\)-algebra \(A\), we prove that the cohomology groups of \(A\)-elliptic complexes of pseudodifferential operators in finitely generated projective \(A\)-Hilbert bundles over compact manifolds are finitely generated \(A\)-modules and Banach spaces provided the images of certain extensions of the so-called associated Laplacians are closed. We also prove that under this condition, the cohomology groups are isomorphic to the kernels of the associated Laplacians. This establishes a Hodge theory for these structures.
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Krýsl, S. Hodge theory for elliptic complexes over unital \(C^*\)-algebras. Ann Glob Anal Geom 45, 197–210 (2014). https://doi.org/10.1007/s10455-013-9394-9
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DOI: https://doi.org/10.1007/s10455-013-9394-9